Laser interferometry has tremendous applications in several fields including manufacturing and biology. One of the serious limitations of this technique is the inherent speckle noise in laser interferometry of rough surfaces. Waves reflected or transmitted by rough surfaces result in unclear interference pattern. Therefore speckle noise reduction is an important step in laser interferometry of the rough surfaces. Many methods have been proposed over the years to compensate the speckle noise in laser interferometric based technologies such as optical coherent tomography, synthetic aperture radar (SAR), ultrasound, etc. One proposed technique to suppress the speckle noise is moving the aperture of the camera. This technique averages the speckle pattern and reduces the effect of noise. The mechanism of moving the aperture increases the complexity of the system which limits the development of this technique. The post-processing methods such as median filter, Wiener filter; enhanced Lee filter, anisotropic diffusion method, etc. have more flexibility and are growing rapidly. These methods improve the signal to noise ratio, but cause the loss of details because of blurring. Fast Fourier Transform (FFT) pass band filters and power spectrum density (PSD) are alternative noise reduction techniques, which can filter off certain level of noise, although they can also distort the fringe pattern. Statistical analysis such as auto-correlation, cross correlation, and Bi-Spectrum are also used to effectively suppress the noise in optical imaging. These techniques are applied in biology, astronomy, under water imaging, and etc., where the noise level is high.
Bi-spectrum is a noise reduction technique in the analysis of nonlinear systems. It is mathematically defined as the Fourier transform of the third order cumulant of a signal:
      B    ⁡          (                        f          1                ,                  f          2                    )        =                    lim        ⁢                  1          T                    <                        X          ⁡                      (                          f              1                        )                          ×                  X          ⁡                      (                          f              2                        )                          ×                              X            *                    ⁡                      (                                          f                1                            +                              f                2                                      )                              >      T        →    ∞  where, X is the Fourier transform of a random process x(t) over a finite time interval T.
Bi-spectrum is a higher-order spectrum and provides supplementary information to the power spectrum. This statistic shows the correlation between the spectral components of the system. One particular property of bi-spectrum is that it retains the phase and the magnitude information of the signal while eliminating the additive Gaussian noise. Bi-spectrum has so many applications in optical processing, geophysics, oceanography, biomedicine, astronomy, etc.
Conventionally, there are several techniques to calculate the phase and magnitude of a signal from its computed bi-spectrum, although all of these techniques require huge amount of computation and are very time consuming. Physically, in order to have a non-zero value for the bi-spectrum, the medium needs to possess a property to produce a frequency component at f1+f2, therefore for linear medium this techniques is not applicable and bi-spectrum does not contain any useful information of the signal. It would be advantageous to determine mechanisms to allow bi-spectrum techniques to apply to linear mediums.